Stability: Routh–Hurwitz & root locus worksheet with answers

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 4s³ + 9s² + 3s + 18. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

A proportional gain K controls the plant G(s) = 1/(s + 5) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −7.

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 8)). The gain is raised until the closed loop is on the edge of stability. Find the frequency ω of the sustained oscillation.

The plant G(s) = (s + 5)/((s − 2)(s + 2)) has an unstable pole at s = 2. A proportional gain K closes a unity-feedback loop around it. Find the smallest gain K_min that makes the closed loop stable.

For L(s) = K/((s + 7)(s² + 2s + 5)), find the angle of departure θ_d of the root locus from the complex pole in the upper half-plane, measured from the positive real axis.

For L(s) = K/((s + 5)(s² + 6s + 13)), find the angle of departure θ_d of the root locus from the complex pole in the upper half-plane, measured from the positive real axis.